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Metamath Proof Explorer

Theorem equsb3

Description: Substitution in an equality. (Contributed by Raph Levien and FL, 4-Dec-2005) Reduce axiom usage. (Revised by Wolf Lammen, 23-Jul-2023)

		Ref	Expression
	Assertion	equsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 )

Proof

Step	Hyp	Ref	Expression
1		equequ1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 = 𝑧 ↔ 𝑤 = 𝑧 ) )
2		equequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑦 = 𝑧 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 )